Download Introduction to Magneto

Introduction to Magneto-Optics
Katsuaki Sato
Department of Applied Physics
Tokyo University of Agriculture & Technology
ISOM2000 Tutorial
CONTENTS
1.
2.
3.
4.
5.
6.
7.
8.
9.
Introduction
Light and Magnetism
What is the Magneto-Optical Effect?
Electromagnetism and Magneto-Optics
Electronic Theory
Measurement of Magneto-Optical Effect
Magneto-Optical Spectra
Recent Advances in Magneto-Optics
Summary
1. Introduction
• Magneto-Optical Effect：Discovered by Faraday on 1845
• Phenomenon：Change of Linear Polarization to Elliptically
Polarized Light Accompanied by Rotation of Principal Axis
• Cause：Difference of Optical Response between LCP and
RCP
• Application：
–
–
–
–
Magneto-Optical Disk
Optical Isolator
Current Sensors
Observation Technique
2. Light and Magnetism
• Light→Magnetism：Photomagentic Effect
– Thermomagnetic Effect：Curie pt. recording→MO disk
– Light-induced Magnetization：ruby, DMS
– Light-induced spin reorientation→Optical motor
• Magnetism→Light：Magneto-Optical Effect
– Shift or splitting of optical absorption line(Zeeman eff.)
– Magnetic resonance：ESR, magneto-plasma effect
– Magneto-optical effect(Faraday, Kerr, Cotton Mouton)
3.What is the Magneto-Optical
Effect?
• MO Effect in Wide Meaning
Any change of optical response induced by
magnetization
• MO Effect in Narrow Meaning
Change of intensity or polarization induced by
magentization
– Faraday effect
– MOKE(Magneto-optical Kerr effect)
– Cotton-Mouton effect
3.1 Faraday & Voigt
Configurations
• (a) Faraday Configuration:
– Magnetization // Light Vector
• (b)Voigt Configuration:
– Magnetization  Light Vector
3.2 Faraday Effect
• MO effect for optical transmission
– Magnetic rotation（Faraday rotation）F
– Magnetic Circular Dichroism（Faraday Ellipticity） F
• Comparison to Natural Optical Rotation
– Faraday Effect is Nonreciprocal (Double rotation for round
trip)
– Natural rotation is Reciprocal (Zero for round trip)
• Verdet Constant
– F=VlH (For paramagnetic and diamagnetic materials）
Illustration of Faraday Effect
For linearly polarized light
incidence,
Rotation of
Principal axis
• Elliptically polarized light
goes out (MCD)
Elliptically
Polarized light
Linearly polarized
light
• With the principal axis
rotated (Magnetic rotation)
3.3 Faraday rotation of magnetic materials
Materials
Fe
rotation
(deg)
3.825･105
Co
1.88･105
546
〃
2
1.11)
Ni
1.3･105
826
120 K
0.27
1.11)
Y3Fe5O12
250
1150
100 K
1.12)
Gd2BiFe5O12
1.01･104
800
RT
1.13)
MnSb
2.8･105
500
〃
1.14)
MnBi
5.0･105
633
〃
1.15)
YFeO3
4.9･103
633
〃
1.16)
NdFeO3
4.72･104
633
〃
1.17)
CrBr3
1.3･105
500
1.5K
1.18)
EuO
5･105
104
660
4.2 K
2.08
1.19)
CdCr2S4
3.8･103
35(80K)
1000
4K
0.6
1.20)
figure of
merit(deg/dB)
44
1.43
wavelength
(nm)
578
temperat
ure
(K)
RT
Mag. field
(T)
2.4
literatu
re
1.11)
3.4 Magneto-Optical Kerr Effect
• Three kinds of MO Kerr effects
– Polar Kerr（Magnetization is oriented
perpendicular to the suraface）
– Longitudinal Kerr（Magnetization is in plane
and is parallel to the plane of incidence）
– Transverse Kerr （Magnetization is in plane
and is perpendicular to the plane of incidence）
3.5 MO Kerr rotation of magnetic materials
rotation
Photon
energy
(deg)
(eV)
temperat
ure
(K)
Fe
0.87
0.75
RT
1.21)
Co
0.85
0.62
〃
1.21)
Ni
0.19
3.1
〃
1.21)
Gd
0.16
4.3
〃
1.22)
Fe3O4
0.32
1
〃
1.23)
MnBi
0.7
1.9
〃
1.24)
PtMnSb
2.0
1.75
〃
1.7
1.8)
CoS2
1.1
0.8
4.2
0.4
1.25)
CrBr3
3.5
2.9
4.2
1.26)
EuO
6
2.1
12
1.27)
USb0.8Te
9.0
0.8
10
0.2 S
CoCr
2 4
4.5
0.7
80
1.29)
a-GdCo
*
CeSb
0.3
1.9
RT
1.30)
2
1.31)
aterials
90
field
literature
(T)
4.0
1.28)
4. Electromagnetism and
Magnetooptics
• Light is the electromagnetic wave.
• Transmission of EM wave：Maxwell equation
• Medium is regareded as continuum→dielectric
permeability tensor
– Effect of Magnetic field→mainly to off-diagonal element
• Eigenequation
• →Complex refractive index：two eigenvalues
eigenfunctions：right and left circularpolarization
– Phase difference between RCP and LCP→rotation
– Amplitude difference →circular dichroism
4.1 Dielectric tensor
~ E
D ε
0
  xx

~   
yx

 zx
 xy
 yy
 zy
 xz 

 yz 
 zz 
  yy

~   C 1~ C    
4
4
xy
 
 zy
 ij  ij  ij
Isotromic media；M//z
Invariant C4 for 90°rotation
around z-axis
 xz
  yx
 xx
 zx
  yz 

 xz 
 zz 
 xx   yy
 yx   xy
  yz   zx   zy  0
  xx
~    
 xy
 0

 xy
 xx
0
0 

0 
 zz 
4.2 MO Equations (1)
Maxwell Equation
Eigenequation
Eigenvalue
~   2
rot rot E  2
E 0
2
c t
 Nˆ 2   xx

  xy

0

  xy
Nˆ 2   xx
0
0  E x 
 
0  E y   0

  zz  E z 
Nˆ 2   xx  i xy
Eigenfunction：LCP and RCP
Without off-diagonal terms：No difference between LCP & RCP
No magnetooptical effect
MO Equations (2)
Nˆ  Nˆ   Nˆ    x x  i x y   x x  i x y  i
xy
 xx
Nˆ 
i  x y
F  




 xx
 (xy1) M
i



 (xx0)  12  (xx2) M 2
Both diagonal and off-diagonal terms contribute to
Magneto-optical effect
4.3 Phenomenology of MO effect
Linearly polarized light can be
decomposed to LCP and RCP
Difference in phase causes rotation of
the direction of Linear polarization
Difference in amplitudes makes
Elliptically polarized light
In general, elliptically polarized light
With the principal axis rotated
5. Electron theory of Magneto-Optics
• Magnetization→Splitting of spin-states
– No direct cause of difference of optical response
between LCP and RCP
• Spin-orbit interaction→Splitting of orbital states
– Absorption of circular polarization→Induction of circular
motion of electrons
• Condition for large magneto-optical response
– Presence of strong (allowed) transitions
– Involving elements with large spin-orbit interaction
– Not directly related with Magnetization
5.1 Microscopic concepts of
electronic polarization
E
+
+
-
Wavefunction
perturbed by
electric field
Unperturbed
wavefunction
+
-
=
+ ・・
+
+
S-like
P-like
Expansion by unperturbed
orbitals
5.2 Orbital angular momentum-selection
rules and circular dichroism
py-orbital
px-orbital
Lz=+1
p+=px+ipy
Lz=-1
p-=px-ipy
Lz=0
s-like
5.3 Role of Spin-Orbit Interaction
Jz=-3/2
Jz=-1/2
L=1
LZ=+1,0,-1
L=0
Without
magnetization
LZ=0
Exchange
splitting
Jz=+1/2
Jz=+3/2
Jz=-1/2
Jz=+1/2
Exchange
+spin-orbit
5.4 MO lineshapes (1)
1.Diamagnetic lineshape
Excited state
”xy
’xy
Lz=-1

0
Lz=+1
1
2
1+2
Ground state
Lz=0
Without
magnetization
With
magnetization
Photon energy
Photon energy
5.4 MO lineshapes (2)
excited state
0
f+
f-
dielectric constant
 f=f+ - f’xy
”xy
ground state
without magnetic
field
with magnetic
field
(a)
photon energy
(b)
6. Measurement of MO effect
1. Cross-polarizer
technique
2. Vibrating polarizer
technique
3. Rotating analyzer
technique
4. Faraday modulation
technique
5. Optical retardation
modulation
6. Measuring system
for MO spectrum
7. Measurement of
elleipticity
6.1 Cross-Nicol technique
P
L
B
A
P
F
(a)
D
S
A
I
P=A+/2
B
/4 rotation
/2 rotation
(b)
 rotation
6.2 Vibrating polarizer
technique
P
+F
B
ID
S
F
D
A
P
6.3 Rotating analyzer
technique
E
B
A=pt
F
ID
D
S
P
A
6.4 Faraday modulation
technique
=0+sin pt
Faraday modulator
B
F
ID
S
P
I=I0+ I sin pt
Zero method
D
A
6.5 Retardation modulation
technique
i
/4
B
j
P
D
PEM
quartz
A
Isotropic
medium
fused silica
CaF2
Ge etc.
Retardation
=(2/)nl sin pt
=0sin pt
l
6.6 Spectral measurement
L
M1
MC
PEM
(p Hz)
C (f Hz)
P
S
M2
LA1 (f Hz)
LA2 (p Hz)
Preamplifier
LA3 (2p Hz)
6.7 Measurement of ellipticity
y
E0sin
E
y’
y

E0

x’
E’
x
x
Optic
axis
E0cos



E  E 0 (cos i  i sin  j )


i


2
E '  E 0 (cos i  i e sin  j )


/4plate
 E 0 cos i  sin  j 

 E0 i '
7. MO spectra of materials
•
•
•
•
•
•
•
Magnetic garnets
Metallic ferromagnet：Fe, Co, Ni
Intermetallic compounds and alloys：PtMnSb etc.
Magnetic semiconductor：CdMnTe etc.
Superlattices：Pt/Co, Fe/Au etc.
Amorphous：TbFeCo, GdFeCo etc.
Granular：Al2O3:Coなど
Theory and experiment of MO
spectra in Fe
Katayama
theory
MO spectra of PtMnSb
K 
カー回転と楕円率
(a)
 xy
 xx 1   xx 
誘電率対角成分
(b)
誘電率非対角成分
(c)
MO spectra in RE-TM (1)
Polar Kerr rotation (min)
Wavelength (nm)
MO spectra in RE-TM(2)
Wavelength (nm)
300
400
500
Polar Kerr rotation (deg)
0
600
700
-0.2
-0.4
-0.6
5
4
3
Photon Energy (eV)
2
Recent Advances in
Magneto-Optics
• Scanning Near Field Magneto-Optical
Microscope (MO-SNOM)
• Nonlinear Magneto-Optics
• Sagnac Magneto-Optical Microscope
• X-ray Magneto-Optical Imaging
SUMMARY
• Basic concept of magneto-optics is
described.
• Macroscopic and microscopic origins of
magneto-optics are described.
• Some of the recent development of
magneto-optics is also given.